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Cauchy problem and initial trace for a doubly degenerate parabolic equation with strongly nonlinear sources. (English) Zbl 0993.35057
Let S T = N ×(0,T), N1 and T>0. The author investigates in S T the Cauchy problem for the equation u t =div(|Du m | p-2 Du m )+u q with the initial condition u(x,0)=u 0 (x). Here p>1, m>0, m(p-1)>1, q>1, and u 0 is locally integrable in N . Of course, for m=1 we have the familiar evolution p-Laplacian equation, and for p=2 we have the porous media equation. The Cauchy problem for the general case is investigated for a large class of initial conditions. To describe this class, the following norm is defined for h1. |||f||| h =sup x N f h (B 1 (x)). Here, · h represents the usual norm in L h (B 1 (x)), where B 1 (x) denotes the unit ball centered at x in N . The first result is the following. Assume u 0 0, |||u 0 ||| h <, where h=1 if q<m(p-1)+p/N and h>(N/p)(q-m(p-1)) otherwise. Then there is a constant T 0 >0 depending on the data such that a solution u(x,t) exists in S T 0 . Quantitative bounds for the solution and results involving a supersolution are obtained. Also the problem of uniqueness is discussed.
MSC:
35K65Parabolic equations of degenerate type
35K15Second order parabolic equations, initial value problems
35K55Nonlinear parabolic equations
35B45A priori estimates for solutions of PDE